Problem 94 - Algebra A Combined Function

Question

Identify the domain and then graph each function. $$ g(x)=\sqrt[3]{x+1} $$

Step-by-Step Solution

Answer

The domain of \( g(x) = \sqrt[3]{x+1} \) is all real numbers. The graph is a cube root curve shifted left by 1 unit.

Step 1: Understanding the Function

The function given is a cube root function. Cube root functions, such as \( g(x) = \sqrt[3]{x+1} \), can have any real number input. This is because the cube root of any real number is valid and defined. Unlike square root functions, we do not have to restrict the domain to non-negative values.

Step 2: Identifying the Domain

Since \( g(x) = \sqrt[3]{x+1} \) is a cube root function, we can conclude that the domain of this function is all real numbers. Mathematically, the domain is expressed as \( (-\infty, \infty) \).

Step 3: Graphing the Function

To graph \( g(x) = \sqrt[3]{x+1} \), we note that it is a transformation of the parent function \( f(x) = \sqrt[3]{x} \). The graph of \( f(x) = \sqrt[3]{x} \) passes through the origin and is symmetric about the origin. The transformation \( x \to x + 1 \) shifts the graph left by 1 unit. So, start by plotting key points such as (-1,0), (0, 1), and (1, 2), and then draw the graph passing through these points, maintaining the same shape as the parent function but shifted left.

Key Concepts

Function Domain

When dealing with functions, particularly cube root functions like \( g(x) = \sqrt[3]{x+1} \), understanding the domain is crucial. The domain of a function is the set of all possible inputs, commonly known as \( x \)-values, that the function can accept. For cube root functions, the domain is quite unrestricted compared to square root functions. This is because cube roots are defined for all real numbers, whether they are positive, negative, or zero.

For example, the cube root of \(-8\) is \(-2\), because \((-2)^3 = -8\). Similarly, the cube root of \(0\) is \(0\), and \(8\) is \(2\) because \(2^3 = 8\). Therefore, the function \( g(x) = \sqrt[3]{x+1} \) has a domain of all real numbers, expressed in interval notation as \((-\infty, \infty)\). This means you can input any real number into this function, and you will get a valid output.

Cube Root Transformation

The concept of transformation refers to changing a function's position, shape, or size on the graph. For the function \( g(x) = \sqrt[3]{x+1} \), the transformation from the base or parent function \( f(x) = \sqrt[3]{x} \) involves shifting the graph horizontally.

A transformation like \( x + 1 \) inside the function modifies where the graph of the original function will appear. Specifically, it shifts the entire graph of \( f(x) = \sqrt[3]{x} \) to the left by 1 unit. This is a horizontal shift because the transformation is applied directly to the \( x \)-variable inside the root function.

In general:

• Adding a positive number to \( x \) inside the function, \( (x+c) \), moves the graph to the left by \( c \) units.
• Adding a negative number to \( x \) inside the function, \( (x-c) \), moves the graph to the right by \( c \) units.

Understanding these transformations helps in quickly sketching the graph, recognizing identical shapes, and accurately predicting the new positions after shifts.

Graphing Techniques

Graphing a function, such as \( g(x) = \sqrt[3]{x+1} \), involves several steps to accurately display its behavior visually. Let’s explore some useful techniques for graphing cube root functions effectively.

First, identify key points based on the transformation of the parent function \( f(x) = \sqrt[3]{x} \). This base graph passes through the origin and is symmetric about the origin. Transformed points are derived from adjusting these key points by the transformations involved. In \( g(x) = \sqrt[3]{x+1} \), the horizontal shift left by 1 modifies key points like:

• (0, 0) becomes (-1, 0).
• (1, 1) becomes (0, 1).
• (-1, -1) becomes (-2, -1).

Second, plot these transformed points on the coordinate axes. These points guide the curve of your graph. The shape remains consistent with that of the parent function, but is now positioned as dictated by the transformation. The graph should smoothly pass through these points to represent the continuous nature of cube root functions.

Finally, familiarize yourself with recognizably curved paths of cube root graphs which differ from the straight lines in linear equations or the parabola shapes in quadratic equations. With these techniques, you'll be able to effectively illustrate the function's behavior on a graph.